Control Design for LPV Systems with Input Saturation and State ...
2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011
Control Design for LPV Systems with Input Saturation and State Constraints: an Application to a Semi-Active Suspension A.L. Do, J.M. Gomes da Silva Jr., O. Sename and L. Dugard Abstract— This paper proposes a control design strategy for LPV systems subject to additive disturbances in the presence of actuator saturation and state constraints. LMI conditions are derived in order to simultaneously compute an LPV controller and an anti-windup gain that ensures the boundedness of the trajectories, considering that the disturbances belong to a given admissible set. The disturbance attenuation is addressed via an H∞ constraint. Besides, state constraints (corresponding to the local validity of the LPV model and system structural limits) are always assured. The theoretical results are applied to a quarter-car model rewritten in the LPV framework where the passivity constraint is recast to the saturation one. The interest of the provided methodology is emphasized by simulations.
I. I NTRODUCTION In the last years, many studies have focused on the control of saturated (in states, control inputs...) systems which are present in almost real applications. For a system with input saturation, there is usually an inconsistency between the states of the plant and those of the controller because of the saturated actuator between the system control input and the controller output. This effect, usually called windup, dramatically degrades the closed-loop performances or even worse causes the system instability. To preserve the consistency, the input to controller needs to be changed by an appropriate signal, which is provided by a called antiwindup compensator. Usually, when a system is subject to actuator saturation, two main issues arise: the guarantee of stability (global or local) and the minimization of the performance degradation. There are two methods to solve these problems: two-step and one-step design. The traditional two-step method first designs a linear controller without considering the input saturation effect and then add an anti-windup compensator to minimize the adverse effects of control input saturation on closed-loop performance [1], [2]. For the one step approach, the controller and an antiwindup compensator (static in general) are simultaneously computed [3], [4]. It can be noticed that the control design with input saturation is a nonlinear problem. However, many solutions have been proposed to model the saturation effect in such a way that the problem can be treated within a linear framework, for example: the polytopic differential inclusion This work was supported by the French national project INOVE/ ANR 2010 BLAN 0308 A.L. Do, O. Sename and L. Dugard are with GIPSA-lab, Control Systems Dept, CNRS-Grenoble INP, ENSE3, BP 46, F-38402 St Martin d’H`eres cedex, France {anh-lam.do, olivier.sename,
luc.dugard}@gipsa-lab.grenoble-inp.fr J.M. Gomes da Silva, Jr. is with the Department of Electrical Engineering, UFRGS, Porto Alegre-RS 90035-190, Brazil [email protected]. This author is also supported by CNPQ, Brazil.
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model [5], [6], [7] and the use of sector conditions [8], [9], [4]. Up to now, numerous results have been obtained for LTI systems. On the other hand, very few papers dealing with switched or LPV systems can be found in the literature, see for instance [10], for switching systems, and [6], [11], [12] for LPV systems. In this paper, we aim at using the one-step anti-windup design for semi-active suspension control to achieve the best compromise among conflicting objectives: passenger comfort, road holding and suspension deflection. Indeed semi-active suspensions have recently received a lot of attention since they provide the best compromise between cost (energy-consumption and actuators/sensors hardware) and performance (see e.g. [13], [14]). For such suspensions, numerous control approaches have been developed. An overview of some recent methodologies in terms of performances is found in [15]. In our previous works [16] and [17], the LPV framework is used to model the nonlinear damper characteristics, and, also to consider the actuator saturation as a scheduling parameter (this approach can be referred to [6]). The performance on suspension deflection, along with comfort and road holding, is managed by using some frequency-based weighting functions. An LPV controller is then synthesized using a global analysis (global stability and performance). In this work, instead of considering the suspension deflection as a performance objective, we will treat it as a constraint. Besides, we are only interested in a certain working range of the damper because, in real applications, its deflection velocity is limited. Since the states are physically bounded, due to the limit in the suspension deflection, and the LPV polytopic model is not globally valid in the state space, a regional stabilization approach is considered. First, a general design method for LPV system with input saturation and state constraints is proposed. Precisely, a sufficient condition to guarantee the regional asymptotic stability of the origin for arbitrary scheduling parameters and to guarantee bounded trajectories in the presence of disturbances (which are assumed to be limited in amplitude) is derived based on the modified sector condition [8] and on the use of a quadratic Lyapunov function. The condition ensures also an upper bound on the induced-L2 gain between the disturbance input and the controlled output when there is no saturation. Moreover, the state constraints on the system are always assured for the considered class of disturbances. Then we apply the result to enhance the performance of a semi-active suspension system rewritten in the LPV framework where the passivity constraint is recast in an input saturation one.
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The rest of the paper is organized as follows. In Section II, we introduce the control problem of LPV system subject to input saturation. In Section III, some useful preliminaries are presented. The main result is stated in Section IV. In Section V, the proposed method is applied to semiactive suspension control. Finally, some conclusions and perspectives are drawn in Section VI. II. P ROBLEM F ORMULATION In the following, Xi denotes the ith row of matrix X. (*) stands for symmetric blocks and sym(X) = X +X T . (•) stands for an element that has no influence on the development.
B. LPV controller We consider a dynamic LPV controller with a static antiwindup action x˙c
= Ac (θ )xc + Bc (θ )uc + Ec (θ )(sat(yc ) − yc )
yc
= Cc (θ )xc + Dc (θ )uc
(7)
where xc ∈ Rnc , uc ∈ R p , yc ∈ Rm and Ec (θ ) is a static antiwindup term [8], [10]. In the presence of the control bounds, the interconnections between the plant and the controller are given (according to (2)) by u = sat(yc ), uc = y, v = Ec (θ )(sat(yc ) − yc )
(8)
where the saturated function sat(.) is defined by A. System description sat(yci ) = sign(yci ) min(|yci |, ui )
Consider a quasi-LPV plant
(9)
From (1) and (2), the closed-loop system is given by x˙ = A(θ )x + Bw (θ )w + Bu u
(1)
ξ˙
z = Cz (θ )x + Dzw (θ )w + Dzu u = Cy x + Dyw w
y
where x ∈ Rn , u ∈ Rm , w ∈ Rq , z ∈ Rr and y ∈ R p are the state, the input, the disturbance vectors, the control output and the measured output, respectively. θ is a vector of scheduling parameters which are supposed to depend on states and assumed to be known (measured or estimated). consider also an LPV controller x˙c
= Ac (θ )xc + Bc (θ )uc + v
yc
= Cc (θ )xc + Dc (θ )uc
where ξ = [xT xcT ]T , ψ(yc ) = yc − sat(yc ) A(θ ) + Bu Dc (θ )Cy BuCc (θ ) A (θ ) = Bc (θ )Cy Ac (θ ) Bw (θ ) + Bu Dc (θ )Dyw B(θ ) = Bc (θ )Dyw Bu 0 Bu = ,R = (11) 0 Inc C (θ ) = Cz (θ ) + Dzu Dc (θ )Cy DzuCc (θ )
(2)
where xc ∈ Rnc , uc ∈ R p , yc ∈ Rm , v is an additional input used for anti-windup compensation. The unconstrained closed-loop system composed by the plant and the controller is defined by the following interconnections u = yc , uc = y, v = 0 (3) The following assumptions are considered: •
Assumption 1: The matrices Bu , Dzu , Cy and Dyz are supposed to be parameter-independent (to satisfy the hypotheses of polytopic design for LPV systems [18]).
•
Assumption 2: The input disturbance is limited in amplitude, that is ∀t > 0, w(t) ∈ W with W = {w ∈ Rq : wT w < δ }
•
•
(4)
Assumption 3: The scheduling parameters depend on the system’s states θ = θ (x,t) and are bounded in Θ = {θ : θ i 6 θi 6 θ i , i = 1, ..., k}
= A (θ )ξ + B(θ )w − (Bu + REc (θ ))ψ(yc ) (10)
z = C (θ )ξ + D(θ )w + Dψ ψ(yc )
(5)
Assumption 4: The control inputs are bounded in amplitude: −ui 6 ui (t) 6 ui , i = 1, ..., m (6)
D(θ ) = Dzw (θ ) + Dzu Dc (θ )Dyw Dψ = −Dzu The controller output is rewritten as yc = K (θ )ξ + Kw (θ )w where K (θ ) = Dc (θ )Cy
(12)
Cc (θ ) , Kw (θ ) = Dc (θ )Dyw
C. Problem Definition In this paper, we look for an LPV controller (7) for the LPV system (1) such that the following conditions are satisfied: (i) in the absence of disturbances, or if the disturbances are vanishing, the controller guarantees the regional asymptotic stability of the origin for an arbitrary scheduling parameter θ provided that the initial states belong to a specific set in the state space. In the presence of disturbances satisfying Assumption 2, the controller guarantees that the trajectories of (10) are bounded. (ii) the controller guarantees the respect of some constraints on the states of the closed-loop system. (iii) for the unconstrained closed-loop system, i.e. when the saturation is not active, the controller guarantees an upper bound γ on the L2 -gain between the disturbance input w and the controlled output z.
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Remark: Considering the same L2 performance when the system operates linearly and under control saturation can lead to very conservative results. Hence, we consider that the L2 performance should be satisfied only by the unconstrained system, which corresponds to a classic H∞ problem. On the other hand, if the control saturates, we should ensure that the trajectories are bounded and do not violate the state constraints. III. P RELIMINARIES A. Practical validity region In practice, besides the constraint on the control input, the system states are usually bounded because of structural limits. Furthermore, the local validity of the LPV model can be also translated in state constraints. We assume the state constraint can be represented by a polyhedron X defined by X = {ξ ∈ R2n : Hi ξ ≤ h0i , i = 1 : s}
(13)
along the trajectories of (10). By using the S-procedure, this condition can be satisfied if there exist scalars β1 > 0 and β2 > 0, such that V˙ + β1 (ξ T Pξ − 1) + β2 (δ − wT w) < 0 IV. M AIN RESULTS In this section, an LMI-based constructive condition to solve the problem stated in II-C is stated. Theorem 1: If, for given β1 > 0 and γ > 0, there exist symmetric positive definite matrices X,Y ∈ Rn×n , a positive scalar β2 , positive diagonal matrices S ∈ Rm×m , matrices ˆ ) ∈ Rn×n , B(θ ˆ ), Zˆ 1 (θ ), Zˆ 2 (θ ) ∈ Rm×n , ˆ ) ∈ Rn×p , C(θ A(θ m×p n×m ˆ ˆ D(θ ) ∈ R , Q(θ ) ∈ R such that the matrix inequalities (20)-(24) are verified, then the LPV controller (2) with matrices
Note that only the state of the plant is constrained, so we have H = H1 0 . B. Saturation model validity region Due to the boundness of w and to the fact that the states of the real system are constrained to belong to X , a regional stabilization approach is adopted in this paper. In order to take into account the saturation effects, an ”LPV” version of the modified sector condition proposed in [8] is applied. With this aim, define the matrix G (θ ) = G1 (θ ) G2 (θ ) and the following polyhedral set Sθ = ξ ∈ R2n , | (Ki (θ ) − Gi (θ )) ξ | 6 ui , i = 1, ..., m , (14) ∀θ ∈ Θ. Hence, the following Lemma can be stated. Lemma 1: If ξ (t) ∈ Sθ , then the following inequality
ψ (yc )T T ψ (yc ) −
G (θ )
0
Kw (θ )
Ec (θ ) = N −1 Qˆ (θ ) S−1 − N −1Y Bu Dc (θ ) = Dˆ (θ ) C (θ ) = [Cˆ (θ ) − D (θ )C X]M −T c c y Bc (θ ) = N −1 [Bˆ (θ ) −Y Bu Dc (θ )] Ac (θ ) = N −1 [Aˆ (θ ) − NBc (θ )Cy X −Y BuCc (θ ) M T −Y (A (θ ) + Bu Dc (θ )Cy ) X]M −T
(19)
where M and N verify MN T = I − XY , solves the problem defined in Section II-C.
ξ ψ (yc ) 6 0 w (15)
holds for any diagonal and positive definite matrix T ∈ Rm×m . Proof: The result can be inferred directly from [8]. C. W-invariance Because the disturbance input is bounded in amplitude, we use the W-invariance concept to ensure the boundedness of the trajectories (see [19]). Definition 1: A set E ⊂ R2n is W-invariant with respect to system (10) if ∀ξ (0) ∈ E , w(t) ∈ W and for any scheduling parameter signal θ (t), it follows that the state trajectory remains in E , i.e ξ (t) ∈ E , ∀t > 0. In the approach, E is considered as an ellipsoidal set associated to a quadratic function V (t) = ξ T Pξ , P = PT 0 E = {ξ ∈ R2n : ξ T Pξ < 1}
(18)
(16)
To ensure that E is a W-invariant set, it suffices to ensure that ∀ξ (t) : ξ T Pξ > 1 ˙ (17) V (t) < 0, ∀w(t) : wT w < δ 3418
L11 (θ ) L12 (θ ) L13 (θ ) L14 (θ ) ∗ L22 (θ ) L23 (θ ) L24 (θ ) ≺0 ∗ ∗ L33 (θ ) L34 (θ ) ∗ ∗ ∗ L44 (θ )
(20)
O11 (θ ) O12 (θ ) O13 (θ ) O14 (θ ) ∗ O22 (θ ) O23 (θ ) O24 (θ ) ≺0 ∗ ∗ O33 (θ ) O34 (θ ) ∗ ∗ ∗ O44 (θ )
(21)
X ∗ ∗ I Y ∗ 0 ˆ ˆ ˆ ˆ Ci (θ ) − Z1i (θ ) (D (θ )Cy )i − Z2i (θ ) u¯2i for i = 1 : m (22)
X ∗ ∗ I Y ∗ 0 H1i X H1i h20i for i = 1 : s
(23)
β2 δ − β1 ≺ 0
(24)
Aˆ (θ ) = NAc (θ ) M T + NBc (θ )Cy X +Y BuCc (θ ) M T +Y (A (θ ) + Bu Dc (θ )Cy ) X ˆ B (θ ) = NBc (θ ) +Y Bu Dc (θ ) Cˆ (θ ) = C (θ ) M T + D (θ )C X
where ˆ ) + C(θ ˆ )T BTu + β1 X L11 (θ ) = A(θ )X + XA(θ )T + BuC(θ ˆ )T + Bu D(θ ˆ )Cy + β1 In L12 (θ ) = A(θ ) + A(θ T ˆ )Dyw + Bw (θ ) L13 (θ ) = −Bu S + Zˆ 1 (θ ) , L14 (θ ) = Bu D(θ
c
ˆ ) + C(θ ˆ )T BTu O11 (θ ) = A(θ )X + XA(θ )T + BuC(θ ˆ )T + A(θ ) + Bu D(θ ˆ )Cy O12 (θ ) = A(θ
(25)
ˆ )Dyw O13 (θ ) = Bw (θ ) + Bu D(θ T ˆ )T DTzu O14 (θ ) = XCz (θ ) + C(θ ˆ )Cy +CyT B(θ ˆ )T O22 (θ ) = YA(θ ) + A(θ )T Y + B(θ ˆ )Dyw O23 (θ ) = Y Bw (θ ) + B(θ ˆ )T DTzu , O33 (θ ) = −γIm O24 (θ ) = Cz (θ )T +CyT D(θ T T ˆ O34 (θ ) = Dzw (θ ) + Dyw D(θ )T DTzu , O44 (θ ) = −γI p
Proof of theorem 1 Sufficient condition for stability - related to problem (i) First, we look for the stability condition for the closedloop system with controller (10). From (15) and (18), by employing the S-procedure, if there exist a positive definite matrix T and positive scalars β1 and β2 such that dV T T − 2ψ(yc )TT× dt + β1 (ξ Pξ − 1) + β2 (δ − w w) ξ ψ (yc ) − G (θ ) 0 Kw (θ ) ψ (yc ) < 0 w (26) then it follows that V˙ < 0, for all ξ in the boundary of E that belongs to the region Sθ , and for all w ∈ W . Hence, in order to ensure that E is a W-invariant set, we must also satisfy:
E ⊂ Sθ
T dV T T dt + β1 ξ Pξ − β2 w w − 2ψ(yc ) T 1 × ξ ψ (yc ) − G (θ ) 0 Kw (θ ) ψ (yc ) < 0 w (28) β2 δ − β1 < 0 (29)
The condition (28) is equivalent to the matrix inequality (33). Note that (33) is not an LMI in terms of β1 , β2 , P, T and the controller matrices Ac , Bc , Cc , Dc , Ec . By first assuming that β1 is known and applying some congruence transformations, similar to the ones proposed in [18], we show in the sequel that (33) is equivalent to (20). With this aim, let P and P−1 be partitioned as follows Y N X M −1 P= and P = (30) NT • MT • and define the matrices X Π= MT
I 0
, S = T −1
y
(32) Pre and post-multiplying (33) by diag(ΠT , S, I) and its transpose, we obtain the LMI (34) (which corresponds exactly to the LMI (20)). On the other hand, it can be seen that the following inequality implies E ⊂ Sθ : P ∗ 0, i = 1, .., m (35) Ki (θ ) − Gi (θ ) u2i Pre and post-multiplying (35) by diag(ΠT , 1), we obtain LMI (22). State constraint - related to problem (ii) To ensure the state constraint (13) is not violated, it suffices to guarantee the inclusion of W-invariant set E in the practical validity region X . Similarly to the previous manipulation, the inequality P ∗ 0 (36) Hi h20i implies that E ⊂ X . Pre and post-multiplying (36) by diag(ΠT , 1) and its transpose, one obtains the LMI (23). Sufficient condition of L2 gain performance in linear mode (without saturation) - related to problem (iii) Consider now V˙ computed with the unconstrained system, i.e. satisfying (3), and the following inequality
(27)
The condition (26) is in fact guaranteed if both following inequalities hold [20]
c
Dˆ (θ ) = Dc (θ ) Zˆ 1 (θ ) = G1 (θ ) X + G2 (θ ) M T Zˆ 2 (θ ) = G1 (θ ) ˆ Q (θ ) = Y Bu S + NEc (θ ) S
ˆ )Cy +CyT B(θ ˆ )T + β1Y L22 (θ ) = YA(θ ) + A(θ ) Y + B(θ T ˆ ) + Zˆ 2 (θ ) , L24 (θ ) = B(θ ˆ )Dyw +Y Bw (θ ) L23 (θ ) = −Q(θ ˆ L33 (θ ) = −2S, L34 (θ ) = D(θ )Dyw , L44 (θ ) = −β2 I T
dV dt
+ 1γ zT z − γwT w < 0
(37)
Following the same steps as in [18], we can show that (21) ensures that (37) is verified. Hence, we can conclude that the L2 gain of the unconstrained system is smaller than γ. V. A PPLICATION TO SEMI - ACTIVE SUSPENSION CONTROL A. Quarter car model Consider a simple quarter vehicle model made up of a sprung mass (ms ) and an unsprung mass (mus ). A spring with the stiffness coefficient ks and a semi-active damper connect these two masses. The wheel tire is modeled by a spring with the stiffness coefficient kt . In this model, zs (respectively zus ) is the vertical position of ms (respectively mus ) and zr is the road profile. It is assumed that the wheel-road contact is ensured. The dynamical equations of a quarter vehicle are given by
(31) 3419
ms z¨s mus z¨us
= −ks zde f − Fdamper
(38)
= ks zde f + Fdamper − kt (zus − zr ) − ct (˙zus − z˙r )
sym(PA (θ )) + β1 P ∗ ∗
ˆ )) + β1 X sym(A(θ )X + BuC(θ ∗ ∗ ∗
−P (Bu + REc (θ )) + G T (θ ) T −2T ∗
ˆ )T + Bu D(θ ˆ )Cy + β1 In A(θ ) + A(θ ˆ )Cy ) + β1Y sym(YA(θ ) + B(θ ∗ ∗
where zde f = zs − zus is the damper deflection (m) (assumed to be measured or estimated), z˙de f = z˙s − z˙us is the deflection velocity (m/s) (can be directly computed from zde f ) and Fdamper , the damper force, is given as follows: Fdamper = c˙zde f
(39)
The passivity constraint of a semi-active damper is 0 6 cmin 6 c 6 cmax
(40)
Rewrite Fdamper = cnom z˙de f + u˙zde f = cnom z˙de f + uθ , where cnom = (cmax + cmin )/2 and θ = z˙de f considered as a scheduling parameter. We suppose that the absolute deflection velocity |˙zde f | is smaller than 1.2 m/s, hence θ is within the range [−1.2, 1.2]. Note that the knowledge of this bound is necessary, when using the polytopic approach. It can be seen that u is the control input and the passivity constraint is now recast into the saturation constraint |u| < (cmax − cmin )/2
The state-space representation of the quarter car model is given by (42)
z = Cz xs + Dz u y
= Cs xs
where xs = (zs − zus , z˙s , zus − zr , z˙us )T , w = z˙r , z = z¨s , y = (zs − zus , z˙s − z˙us )T . 0 1 0 −1 −k −c c h i nom nom ms 0 ms ms s , Dz = −θ As = ms 0 0 0 1 ks mus
cnom mus
−kt mus ct mus
−cnom −ct
i mus h θ 0 Bs1 = 0 0 −1 , Bs2 = 0 −θ mus ms h i 1 0 0 0 cnom −ks −cnom 0 Cz = ms , C = . s ms ms 0 1 0 −1 Note that the input matrices Bs2 and Dz are parameter dependent so the Assumption 1 is not guaranteed. Adding
ˆ )Dyw + Bw (θ ) Bu D(θ ˆ )Dyw +Y Bw (θ ) B(θ ≺0 ˆ )Dyw D(θ −β2 I
(34)
a strict low-pass filter on the control input as in [16], the system can be represented in such a way that Assumption 1 is satisfied. In this preliminary study, the only state constraint is the suspension deflection constraint |zde f | < 0.125 m (because we suppose that the bound of θ is guaranteed during the work of the damper). Hence, in (13), H = 1 0 0 0 and h0 = 0.125. We aim at improving the passenger comfort by minimizing the disturbance attenuation level γ of the closed-loop transfer function from w (the road disturbance) to z (the car acceleration z¨s ) (while taking into account the constraints on the system input and states). To enhance the performance, the weighting function on z is chosen (using the optimization procedure in [21]) Wz (s) =
0.4901s2 + 1563s + 360.9 s2 + 217.7s + 788.9
(43)
The augmented system (42)-(43) is written in the form of (1) and is used for the controller synthesis. C. Simulation Results A common example of road disturbance is described by A 0 ≤ t ≤ VL ± 2 1 − cos 2πV L t , zr = 0, t > VL
B. State-space representation and control objective
= As xs + Bs1 w + Bs2 u
−Bu S + Zˆ 1 (θ )T ˆ ) + Zˆ 2 (θ )T −Q(θ −2S ∗
(33)
(41)
The quarter vehicle used in this paper is the “Renault M´egane Coup´e” model whose specific parameters are: ms = 315 kg, mus = 37.5 kg, ks = 29500 N/m, kt = 210000 N/m, ct =100 Ns/m. The damping coefficient varies between cmin = 700 Ns/m and cmax = 5000 Ns/m. The maximum suspension deflection is 0.125 m (which corresponds to the state constraint).
x˙s
PB (θ ) T Kw (θ ) < 0 −β2 I
where A and L the height and length, V the vehicle velocity, “+” a bump, “-” a pothole. As in figure 1, we consider a road profile with a bump where A = 0.15 m, L = 5 m, V = 27 km/h at 0 s (corresponding to a low frequency disturbance) and a pothole where A = 0.055 m, L = 5 m, V = 72 km/h at 2.5 s (corresponding to a high frequency disturbance). The chosen road profile corresponds to a disturbance satisfying Assumption 2, with δ = 0.5 m2 /s2 . As seen in Fig. 2-3, we can improve the passenger comfort (by minimizing the peak value of the car body acceleration) of the closed-loop system with the proposed method w.r.t the passive open-loop cases (Soft Damper (c = cmin ), Hard Damper (c = cmax ) and Nominal Damper (c = cnom )). Observe that between 2.5 s and 3 s, the control effectively saturates, but the stability is kept. Indeed, during the saturation, the anti-windup acts and the performance does not degrades. Furthermore, it should be noticed that the limits of the suspension travel and the validity for the LPV system are not violated by the trajectory.
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R EFERENCES
Road Profile [m]
0.2
0.1
0
−0.1 0
0.5
1
1.5
2
2.5
3
3.5
4
Road Profile Velocity [m/s]
Time [s] 1 Bound on disturbance amplitude
0.5 0 −0.5
Bound on disturbance amplitude
−1 0
0.5
1
1.5
2
2.5
3
3.5
4
Time [s]
Car Acceleration [m/s2]
Fig. 1.
10
Soft Damper Hard Damper Nominal Damper Proposed Method
5 0 −5 0
Suspension Deflection [m]
Road Profile.
0.5
1
1.5
2
2.5
3
3.5
4
Suspension Deflection Limit (State Constraint)
0.1 0.05 0 −0.05 −0.1
Suspension Deflection Limit (State Constraint)
0
0.5
1
Fig. 2.
1.5
2
Time [s]
2.5
3
3.5
4
Performances Comparison.
θ = z˙def [m/s]
2 Bound on scheduling parameter
1 0 −1
Bound on scheduling parameter
−2 0
0.5
1
1.5
2
2.5
3
3.5
4
Damping coefficient c [Ns/m]
Time [s] 6000
cmax
Saturation bound
Proposed Method
4000 cnom 2000 cmin 0 0
Saturation bound
0.5
1
1.5
2
2.5
3
3.5
4
Time [s]
Fig. 3.
Scheduling parameter and saturation constraint.
VI. C ONCLUSIONS The contribution of the paper is twofold: the proposition of an LMI method to synthesize LPV controllers taking into account input saturation and state constraints; and the application of the method in a semi-active suspension control problem. The simulation results have shown the efficiency of the proposed methodology w.r.t several passive cases. For future work, the application for semi-active suspension control will also be extended further i.e the optimization of comfort, road holding and suspension deflection.
[1] M. V. Kothare, P. J. Campo, M. Morari, and C. N. Nett, “A unified framework for the study of anti-windup designs,” Automatica, vol. 30, pp. 1869–1883, 1994. [2] G. Grimm, J. Hatfield, I. Postlethwaite, A. Teel, M. Turner, and L. Zaccarian, “Antiwindup for stable linear systems with input saturation: An LMI-based synthesis,” IEEE Transaction on Automatic Control, vol. 48, no. 9, pp. 1509–1525, september 2003. [3] J. M. Gomes da Silva Jr., D. Limon, T. Alamo, and E. F. Camacho, “Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints,” IEEE Transaction on Automatic Control, vol. 53, no. 10, pp. 2367–2372, 2008. [4] E. F. Mulder, P. Y. Tiwari, and M. V. Kothare, “Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization,” Automatica, vol. 45, pp. 805–811, 2009. [5] J. M. Gomes da Silva Jr. and S. Tarbouriech, “Local stabilization of discrete-time linear systems with saturating controls: An lmi-based approach,” IEEE Transaction on Automatic Control, vol. 46, no. 1, pp. 119–125, 2001. [6] F. Wu, K. M. Grigoriadis, and A. Packard, “Anti-windup controller design using linear parameter-varying control methods,” International Journal of Control, vol. 73, no. 12, pp. 1104–1114, 2000. [7] T. Hu, Z. Lin, and B. M. Chen, “Analysis and design for discretetime linear systems subject to actuator saturation,” System & Control Letters, vol. 45, pp. 97–112, 2002. [8] J. M. Gomes da Silva Jr. and S. Tarbouriech, “Antiwindup design with guaranteed regions of stability: An lmi-based approach,” IEEE Transaction on Automatic Control, vol. 50, no. 1, pp. 106–111, 2005. [9] F. Wu and B. Lu, “Anti-windup control design for exponentially unstable LTI systems with actuator saturation,” System & Control Letters, vol. 52, pp. 305–322, 2004. [10] V. F. Montagner, J. M. Gomes da Silva Jr., and P. L. D. Peres, “Regional stabilization of switched systems subject to input saturation,” in Proceedings of the 3rd IFAC Symposium on System Structure and Control (SSSC), 2007. [11] V. F. Montagner, R. C. L. F. Oliveira, P. L. D. Peres, S. Tarbouriech, and I. Queinnec, “Gain-scheduled controllers for linear parametervarying systems with saturating actuators: Lmi-based design,” in Proceedings of the IEEE American Control Conference (ACC), New York City, USA, July 2007, pp. 6067–6072. [12] Y.-Y. Cao, Z. Lin, and Y. Shamash, “Set invariance analysis and gainscheduling control for lpv systems subject to actuator saturation,” in Proceedings of the IEEE American Control Conference (ACC), Anchorage, AK, May 2002, pp. 668–673. [13] T. D. Gillespie, Fundamentals of Vehicle Dynamics. Society of Automotive Engineers Inc, 1992. [14] U. Kiencke and L. Nielsen, Automotive Control Systems for Engine, Driveline, and Vehicle. Springer Verlag, 2000. [15] S. Savaresi, C. Poussot-Vassal, C. Spelta, O. Sename, and L. Dugard, Semi-Active Suspension Control for Vehicles. Elsevier, 2010. [16] A. L. Do, O. Sename, and L. Dugard, “An LPV control approach for semi-active suspension control with actuator constraints,” in Proceedings of the IEEE American Control Conference (ACC), Baltimore, Maryland, USA, June 2010. [17] A. L. Do, C. Spelta, S. Savaresi, O. Sename, L. Dugard, and D. Delvecchio, “An LPV control approach for comfort and suspension travel improvements of semi-active suspension systems,” in Proceedings of the 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, December 2010, pp. 5660–5665. [18] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective outputfeedback control via LMI optimization,” IEEE Transaction on Automatic Control, vol. 42, no. 7, pp. 896–911, july 1997. [19] F. Blanchini, “Set invariance in control,” Automatica, vol. 35, no. 11, pp. 1747–1767, 1999. [20] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics, 1994. [21] A. L. Do, O. Sename, L. Dugard, and B. Soualmi, “Multi-objective optimization by genetic algorithms in H∞ /LPV control of semi-active suspension,” in Proceedings of the 18th IFAC World Congress (WC), Milan, Italy, September 2011.
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Control Design for LPV Systems with Input Saturation and State Constraints: an Application to a Semi-Active Suspension A.L. Do, J.M. Gomes da Silva Jr., O. Sename and L. Dugard Abstract— This paper proposes a control design strategy for LPV systems subject to additive disturbances in the presence of actuator saturation and state constraints. LMI conditions are derived in order to simultaneously compute an LPV controller and an anti-windup gain that ensures the boundedness of the trajectories, considering that the disturbances belong to a given admissible set. The disturbance attenuation is addressed via an H∞ constraint. Besides, state constraints (corresponding to the local validity of the LPV model and system structural limits) are always assured. The theoretical results are applied to a quarter-car model rewritten in the LPV framework where the passivity constraint is recast to the saturation one. The interest of the provided methodology is emphasized by simulations.
I. I NTRODUCTION In the last years, many studies have focused on the control of saturated (in states, control inputs...) systems which are present in almost real applications. For a system with input saturation, there is usually an inconsistency between the states of the plant and those of the controller because of the saturated actuator between the system control input and the controller output. This effect, usually called windup, dramatically degrades the closed-loop performances or even worse causes the system instability. To preserve the consistency, the input to controller needs to be changed by an appropriate signal, which is provided by a called antiwindup compensator. Usually, when a system is subject to actuator saturation, two main issues arise: the guarantee of stability (global or local) and the minimization of the performance degradation. There are two methods to solve these problems: two-step and one-step design. The traditional two-step method first designs a linear controller without considering the input saturation effect and then add an anti-windup compensator to minimize the adverse effects of control input saturation on closed-loop performance [1], [2]. For the one step approach, the controller and an antiwindup compensator (static in general) are simultaneously computed [3], [4]. It can be noticed that the control design with input saturation is a nonlinear problem. However, many solutions have been proposed to model the saturation effect in such a way that the problem can be treated within a linear framework, for example: the polytopic differential inclusion This work was supported by the French national project INOVE/ ANR 2010 BLAN 0308 A.L. Do, O. Sename and L. Dugard are with GIPSA-lab, Control Systems Dept, CNRS-Grenoble INP, ENSE3, BP 46, F-38402 St Martin d’H`eres cedex, France {anh-lam.do, olivier.sename,
luc.dugard}@gipsa-lab.grenoble-inp.fr J.M. Gomes da Silva, Jr. is with the Department of Electrical Engineering, UFRGS, Porto Alegre-RS 90035-190, Brazil [email protected]. This author is also supported by CNPQ, Brazil.
978-1-61284-799-3/11/$26.00 ©2011 IEEE
model [5], [6], [7] and the use of sector conditions [8], [9], [4]. Up to now, numerous results have been obtained for LTI systems. On the other hand, very few papers dealing with switched or LPV systems can be found in the literature, see for instance [10], for switching systems, and [6], [11], [12] for LPV systems. In this paper, we aim at using the one-step anti-windup design for semi-active suspension control to achieve the best compromise among conflicting objectives: passenger comfort, road holding and suspension deflection. Indeed semi-active suspensions have recently received a lot of attention since they provide the best compromise between cost (energy-consumption and actuators/sensors hardware) and performance (see e.g. [13], [14]). For such suspensions, numerous control approaches have been developed. An overview of some recent methodologies in terms of performances is found in [15]. In our previous works [16] and [17], the LPV framework is used to model the nonlinear damper characteristics, and, also to consider the actuator saturation as a scheduling parameter (this approach can be referred to [6]). The performance on suspension deflection, along with comfort and road holding, is managed by using some frequency-based weighting functions. An LPV controller is then synthesized using a global analysis (global stability and performance). In this work, instead of considering the suspension deflection as a performance objective, we will treat it as a constraint. Besides, we are only interested in a certain working range of the damper because, in real applications, its deflection velocity is limited. Since the states are physically bounded, due to the limit in the suspension deflection, and the LPV polytopic model is not globally valid in the state space, a regional stabilization approach is considered. First, a general design method for LPV system with input saturation and state constraints is proposed. Precisely, a sufficient condition to guarantee the regional asymptotic stability of the origin for arbitrary scheduling parameters and to guarantee bounded trajectories in the presence of disturbances (which are assumed to be limited in amplitude) is derived based on the modified sector condition [8] and on the use of a quadratic Lyapunov function. The condition ensures also an upper bound on the induced-L2 gain between the disturbance input and the controlled output when there is no saturation. Moreover, the state constraints on the system are always assured for the considered class of disturbances. Then we apply the result to enhance the performance of a semi-active suspension system rewritten in the LPV framework where the passivity constraint is recast in an input saturation one.
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The rest of the paper is organized as follows. In Section II, we introduce the control problem of LPV system subject to input saturation. In Section III, some useful preliminaries are presented. The main result is stated in Section IV. In Section V, the proposed method is applied to semiactive suspension control. Finally, some conclusions and perspectives are drawn in Section VI. II. P ROBLEM F ORMULATION In the following, Xi denotes the ith row of matrix X. (*) stands for symmetric blocks and sym(X) = X +X T . (•) stands for an element that has no influence on the development.
B. LPV controller We consider a dynamic LPV controller with a static antiwindup action x˙c
= Ac (θ )xc + Bc (θ )uc + Ec (θ )(sat(yc ) − yc )
yc
= Cc (θ )xc + Dc (θ )uc
(7)
where xc ∈ Rnc , uc ∈ R p , yc ∈ Rm and Ec (θ ) is a static antiwindup term [8], [10]. In the presence of the control bounds, the interconnections between the plant and the controller are given (according to (2)) by u = sat(yc ), uc = y, v = Ec (θ )(sat(yc ) − yc )
(8)
where the saturated function sat(.) is defined by A. System description sat(yci ) = sign(yci ) min(|yci |, ui )
Consider a quasi-LPV plant
(9)
From (1) and (2), the closed-loop system is given by x˙ = A(θ )x + Bw (θ )w + Bu u
(1)
ξ˙
z = Cz (θ )x + Dzw (θ )w + Dzu u = Cy x + Dyw w
y
where x ∈ Rn , u ∈ Rm , w ∈ Rq , z ∈ Rr and y ∈ R p are the state, the input, the disturbance vectors, the control output and the measured output, respectively. θ is a vector of scheduling parameters which are supposed to depend on states and assumed to be known (measured or estimated). consider also an LPV controller x˙c
= Ac (θ )xc + Bc (θ )uc + v
yc
= Cc (θ )xc + Dc (θ )uc
where ξ = [xT xcT ]T , ψ(yc ) = yc − sat(yc ) A(θ ) + Bu Dc (θ )Cy BuCc (θ ) A (θ ) = Bc (θ )Cy Ac (θ ) Bw (θ ) + Bu Dc (θ )Dyw B(θ ) = Bc (θ )Dyw Bu 0 Bu = ,R = (11) 0 Inc C (θ ) = Cz (θ ) + Dzu Dc (θ )Cy DzuCc (θ )
(2)
where xc ∈ Rnc , uc ∈ R p , yc ∈ Rm , v is an additional input used for anti-windup compensation. The unconstrained closed-loop system composed by the plant and the controller is defined by the following interconnections u = yc , uc = y, v = 0 (3) The following assumptions are considered: •
Assumption 1: The matrices Bu , Dzu , Cy and Dyz are supposed to be parameter-independent (to satisfy the hypotheses of polytopic design for LPV systems [18]).
•
Assumption 2: The input disturbance is limited in amplitude, that is ∀t > 0, w(t) ∈ W with W = {w ∈ Rq : wT w < δ }
•
•
(4)
Assumption 3: The scheduling parameters depend on the system’s states θ = θ (x,t) and are bounded in Θ = {θ : θ i 6 θi 6 θ i , i = 1, ..., k}
= A (θ )ξ + B(θ )w − (Bu + REc (θ ))ψ(yc ) (10)
z = C (θ )ξ + D(θ )w + Dψ ψ(yc )
(5)
Assumption 4: The control inputs are bounded in amplitude: −ui 6 ui (t) 6 ui , i = 1, ..., m (6)
D(θ ) = Dzw (θ ) + Dzu Dc (θ )Dyw Dψ = −Dzu The controller output is rewritten as yc = K (θ )ξ + Kw (θ )w where K (θ ) = Dc (θ )Cy
(12)
Cc (θ ) , Kw (θ ) = Dc (θ )Dyw
C. Problem Definition In this paper, we look for an LPV controller (7) for the LPV system (1) such that the following conditions are satisfied: (i) in the absence of disturbances, or if the disturbances are vanishing, the controller guarantees the regional asymptotic stability of the origin for an arbitrary scheduling parameter θ provided that the initial states belong to a specific set in the state space. In the presence of disturbances satisfying Assumption 2, the controller guarantees that the trajectories of (10) are bounded. (ii) the controller guarantees the respect of some constraints on the states of the closed-loop system. (iii) for the unconstrained closed-loop system, i.e. when the saturation is not active, the controller guarantees an upper bound γ on the L2 -gain between the disturbance input w and the controlled output z.
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Remark: Considering the same L2 performance when the system operates linearly and under control saturation can lead to very conservative results. Hence, we consider that the L2 performance should be satisfied only by the unconstrained system, which corresponds to a classic H∞ problem. On the other hand, if the control saturates, we should ensure that the trajectories are bounded and do not violate the state constraints. III. P RELIMINARIES A. Practical validity region In practice, besides the constraint on the control input, the system states are usually bounded because of structural limits. Furthermore, the local validity of the LPV model can be also translated in state constraints. We assume the state constraint can be represented by a polyhedron X defined by X = {ξ ∈ R2n : Hi ξ ≤ h0i , i = 1 : s}
(13)
along the trajectories of (10). By using the S-procedure, this condition can be satisfied if there exist scalars β1 > 0 and β2 > 0, such that V˙ + β1 (ξ T Pξ − 1) + β2 (δ − wT w) < 0 IV. M AIN RESULTS In this section, an LMI-based constructive condition to solve the problem stated in II-C is stated. Theorem 1: If, for given β1 > 0 and γ > 0, there exist symmetric positive definite matrices X,Y ∈ Rn×n , a positive scalar β2 , positive diagonal matrices S ∈ Rm×m , matrices ˆ ) ∈ Rn×n , B(θ ˆ ), Zˆ 1 (θ ), Zˆ 2 (θ ) ∈ Rm×n , ˆ ) ∈ Rn×p , C(θ A(θ m×p n×m ˆ ˆ D(θ ) ∈ R , Q(θ ) ∈ R such that the matrix inequalities (20)-(24) are verified, then the LPV controller (2) with matrices
Note that only the state of the plant is constrained, so we have H = H1 0 . B. Saturation model validity region Due to the boundness of w and to the fact that the states of the real system are constrained to belong to X , a regional stabilization approach is adopted in this paper. In order to take into account the saturation effects, an ”LPV” version of the modified sector condition proposed in [8] is applied. With this aim, define the matrix G (θ ) = G1 (θ ) G2 (θ ) and the following polyhedral set Sθ = ξ ∈ R2n , | (Ki (θ ) − Gi (θ )) ξ | 6 ui , i = 1, ..., m , (14) ∀θ ∈ Θ. Hence, the following Lemma can be stated. Lemma 1: If ξ (t) ∈ Sθ , then the following inequality
ψ (yc )T T ψ (yc ) −
G (θ )
0
Kw (θ )
Ec (θ ) = N −1 Qˆ (θ ) S−1 − N −1Y Bu Dc (θ ) = Dˆ (θ ) C (θ ) = [Cˆ (θ ) − D (θ )C X]M −T c c y Bc (θ ) = N −1 [Bˆ (θ ) −Y Bu Dc (θ )] Ac (θ ) = N −1 [Aˆ (θ ) − NBc (θ )Cy X −Y BuCc (θ ) M T −Y (A (θ ) + Bu Dc (θ )Cy ) X]M −T
(19)
where M and N verify MN T = I − XY , solves the problem defined in Section II-C.
ξ ψ (yc ) 6 0 w (15)
holds for any diagonal and positive definite matrix T ∈ Rm×m . Proof: The result can be inferred directly from [8]. C. W-invariance Because the disturbance input is bounded in amplitude, we use the W-invariance concept to ensure the boundedness of the trajectories (see [19]). Definition 1: A set E ⊂ R2n is W-invariant with respect to system (10) if ∀ξ (0) ∈ E , w(t) ∈ W and for any scheduling parameter signal θ (t), it follows that the state trajectory remains in E , i.e ξ (t) ∈ E , ∀t > 0. In the approach, E is considered as an ellipsoidal set associated to a quadratic function V (t) = ξ T Pξ , P = PT 0 E = {ξ ∈ R2n : ξ T Pξ < 1}
(18)
(16)
To ensure that E is a W-invariant set, it suffices to ensure that ∀ξ (t) : ξ T Pξ > 1 ˙ (17) V (t) < 0, ∀w(t) : wT w < δ 3418
L11 (θ ) L12 (θ ) L13 (θ ) L14 (θ ) ∗ L22 (θ ) L23 (θ ) L24 (θ ) ≺0 ∗ ∗ L33 (θ ) L34 (θ ) ∗ ∗ ∗ L44 (θ )
(20)
O11 (θ ) O12 (θ ) O13 (θ ) O14 (θ ) ∗ O22 (θ ) O23 (θ ) O24 (θ ) ≺0 ∗ ∗ O33 (θ ) O34 (θ ) ∗ ∗ ∗ O44 (θ )
(21)
X ∗ ∗ I Y ∗ 0 ˆ ˆ ˆ ˆ Ci (θ ) − Z1i (θ ) (D (θ )Cy )i − Z2i (θ ) u¯2i for i = 1 : m (22)
X ∗ ∗ I Y ∗ 0 H1i X H1i h20i for i = 1 : s
(23)
β2 δ − β1 ≺ 0
(24)
Aˆ (θ ) = NAc (θ ) M T + NBc (θ )Cy X +Y BuCc (θ ) M T +Y (A (θ ) + Bu Dc (θ )Cy ) X ˆ B (θ ) = NBc (θ ) +Y Bu Dc (θ ) Cˆ (θ ) = C (θ ) M T + D (θ )C X
where ˆ ) + C(θ ˆ )T BTu + β1 X L11 (θ ) = A(θ )X + XA(θ )T + BuC(θ ˆ )T + Bu D(θ ˆ )Cy + β1 In L12 (θ ) = A(θ ) + A(θ T ˆ )Dyw + Bw (θ ) L13 (θ ) = −Bu S + Zˆ 1 (θ ) , L14 (θ ) = Bu D(θ
c
ˆ ) + C(θ ˆ )T BTu O11 (θ ) = A(θ )X + XA(θ )T + BuC(θ ˆ )T + A(θ ) + Bu D(θ ˆ )Cy O12 (θ ) = A(θ
(25)
ˆ )Dyw O13 (θ ) = Bw (θ ) + Bu D(θ T ˆ )T DTzu O14 (θ ) = XCz (θ ) + C(θ ˆ )Cy +CyT B(θ ˆ )T O22 (θ ) = YA(θ ) + A(θ )T Y + B(θ ˆ )Dyw O23 (θ ) = Y Bw (θ ) + B(θ ˆ )T DTzu , O33 (θ ) = −γIm O24 (θ ) = Cz (θ )T +CyT D(θ T T ˆ O34 (θ ) = Dzw (θ ) + Dyw D(θ )T DTzu , O44 (θ ) = −γI p
Proof of theorem 1 Sufficient condition for stability - related to problem (i) First, we look for the stability condition for the closedloop system with controller (10). From (15) and (18), by employing the S-procedure, if there exist a positive definite matrix T and positive scalars β1 and β2 such that dV T T − 2ψ(yc )TT× dt + β1 (ξ Pξ − 1) + β2 (δ − w w) ξ ψ (yc ) − G (θ ) 0 Kw (θ ) ψ (yc ) < 0 w (26) then it follows that V˙ < 0, for all ξ in the boundary of E that belongs to the region Sθ , and for all w ∈ W . Hence, in order to ensure that E is a W-invariant set, we must also satisfy:
E ⊂ Sθ
T dV T T dt + β1 ξ Pξ − β2 w w − 2ψ(yc ) T 1 × ξ ψ (yc ) − G (θ ) 0 Kw (θ ) ψ (yc ) < 0 w (28) β2 δ − β1 < 0 (29)
The condition (28) is equivalent to the matrix inequality (33). Note that (33) is not an LMI in terms of β1 , β2 , P, T and the controller matrices Ac , Bc , Cc , Dc , Ec . By first assuming that β1 is known and applying some congruence transformations, similar to the ones proposed in [18], we show in the sequel that (33) is equivalent to (20). With this aim, let P and P−1 be partitioned as follows Y N X M −1 P= and P = (30) NT • MT • and define the matrices X Π= MT
I 0
, S = T −1
y
(32) Pre and post-multiplying (33) by diag(ΠT , S, I) and its transpose, we obtain the LMI (34) (which corresponds exactly to the LMI (20)). On the other hand, it can be seen that the following inequality implies E ⊂ Sθ : P ∗ 0, i = 1, .., m (35) Ki (θ ) − Gi (θ ) u2i Pre and post-multiplying (35) by diag(ΠT , 1), we obtain LMI (22). State constraint - related to problem (ii) To ensure the state constraint (13) is not violated, it suffices to guarantee the inclusion of W-invariant set E in the practical validity region X . Similarly to the previous manipulation, the inequality P ∗ 0 (36) Hi h20i implies that E ⊂ X . Pre and post-multiplying (36) by diag(ΠT , 1) and its transpose, one obtains the LMI (23). Sufficient condition of L2 gain performance in linear mode (without saturation) - related to problem (iii) Consider now V˙ computed with the unconstrained system, i.e. satisfying (3), and the following inequality
(27)
The condition (26) is in fact guaranteed if both following inequalities hold [20]
c
Dˆ (θ ) = Dc (θ ) Zˆ 1 (θ ) = G1 (θ ) X + G2 (θ ) M T Zˆ 2 (θ ) = G1 (θ ) ˆ Q (θ ) = Y Bu S + NEc (θ ) S
ˆ )Cy +CyT B(θ ˆ )T + β1Y L22 (θ ) = YA(θ ) + A(θ ) Y + B(θ T ˆ ) + Zˆ 2 (θ ) , L24 (θ ) = B(θ ˆ )Dyw +Y Bw (θ ) L23 (θ ) = −Q(θ ˆ L33 (θ ) = −2S, L34 (θ ) = D(θ )Dyw , L44 (θ ) = −β2 I T
dV dt
+ 1γ zT z − γwT w < 0
(37)
Following the same steps as in [18], we can show that (21) ensures that (37) is verified. Hence, we can conclude that the L2 gain of the unconstrained system is smaller than γ. V. A PPLICATION TO SEMI - ACTIVE SUSPENSION CONTROL A. Quarter car model Consider a simple quarter vehicle model made up of a sprung mass (ms ) and an unsprung mass (mus ). A spring with the stiffness coefficient ks and a semi-active damper connect these two masses. The wheel tire is modeled by a spring with the stiffness coefficient kt . In this model, zs (respectively zus ) is the vertical position of ms (respectively mus ) and zr is the road profile. It is assumed that the wheel-road contact is ensured. The dynamical equations of a quarter vehicle are given by
(31) 3419
ms z¨s mus z¨us
= −ks zde f − Fdamper
(38)
= ks zde f + Fdamper − kt (zus − zr ) − ct (˙zus − z˙r )
sym(PA (θ )) + β1 P ∗ ∗
ˆ )) + β1 X sym(A(θ )X + BuC(θ ∗ ∗ ∗
−P (Bu + REc (θ )) + G T (θ ) T −2T ∗
ˆ )T + Bu D(θ ˆ )Cy + β1 In A(θ ) + A(θ ˆ )Cy ) + β1Y sym(YA(θ ) + B(θ ∗ ∗
where zde f = zs − zus is the damper deflection (m) (assumed to be measured or estimated), z˙de f = z˙s − z˙us is the deflection velocity (m/s) (can be directly computed from zde f ) and Fdamper , the damper force, is given as follows: Fdamper = c˙zde f
(39)
The passivity constraint of a semi-active damper is 0 6 cmin 6 c 6 cmax
(40)
Rewrite Fdamper = cnom z˙de f + u˙zde f = cnom z˙de f + uθ , where cnom = (cmax + cmin )/2 and θ = z˙de f considered as a scheduling parameter. We suppose that the absolute deflection velocity |˙zde f | is smaller than 1.2 m/s, hence θ is within the range [−1.2, 1.2]. Note that the knowledge of this bound is necessary, when using the polytopic approach. It can be seen that u is the control input and the passivity constraint is now recast into the saturation constraint |u| < (cmax − cmin )/2
The state-space representation of the quarter car model is given by (42)
z = Cz xs + Dz u y
= Cs xs
where xs = (zs − zus , z˙s , zus − zr , z˙us )T , w = z˙r , z = z¨s , y = (zs − zus , z˙s − z˙us )T . 0 1 0 −1 −k −c c h i nom nom ms 0 ms ms s , Dz = −θ As = ms 0 0 0 1 ks mus
cnom mus
−kt mus ct mus
−cnom −ct
i mus h θ 0 Bs1 = 0 0 −1 , Bs2 = 0 −θ mus ms h i 1 0 0 0 cnom −ks −cnom 0 Cz = ms , C = . s ms ms 0 1 0 −1 Note that the input matrices Bs2 and Dz are parameter dependent so the Assumption 1 is not guaranteed. Adding
ˆ )Dyw + Bw (θ ) Bu D(θ ˆ )Dyw +Y Bw (θ ) B(θ ≺0 ˆ )Dyw D(θ −β2 I
(34)
a strict low-pass filter on the control input as in [16], the system can be represented in such a way that Assumption 1 is satisfied. In this preliminary study, the only state constraint is the suspension deflection constraint |zde f | < 0.125 m (because we suppose that the bound of θ is guaranteed during the work of the damper). Hence, in (13), H = 1 0 0 0 and h0 = 0.125. We aim at improving the passenger comfort by minimizing the disturbance attenuation level γ of the closed-loop transfer function from w (the road disturbance) to z (the car acceleration z¨s ) (while taking into account the constraints on the system input and states). To enhance the performance, the weighting function on z is chosen (using the optimization procedure in [21]) Wz (s) =
0.4901s2 + 1563s + 360.9 s2 + 217.7s + 788.9
(43)
The augmented system (42)-(43) is written in the form of (1) and is used for the controller synthesis. C. Simulation Results A common example of road disturbance is described by A 0 ≤ t ≤ VL ± 2 1 − cos 2πV L t , zr = 0, t > VL
B. State-space representation and control objective
= As xs + Bs1 w + Bs2 u
−Bu S + Zˆ 1 (θ )T ˆ ) + Zˆ 2 (θ )T −Q(θ −2S ∗
(33)
(41)
The quarter vehicle used in this paper is the “Renault M´egane Coup´e” model whose specific parameters are: ms = 315 kg, mus = 37.5 kg, ks = 29500 N/m, kt = 210000 N/m, ct =100 Ns/m. The damping coefficient varies between cmin = 700 Ns/m and cmax = 5000 Ns/m. The maximum suspension deflection is 0.125 m (which corresponds to the state constraint).
x˙s
PB (θ ) T Kw (θ ) < 0 −β2 I
where A and L the height and length, V the vehicle velocity, “+” a bump, “-” a pothole. As in figure 1, we consider a road profile with a bump where A = 0.15 m, L = 5 m, V = 27 km/h at 0 s (corresponding to a low frequency disturbance) and a pothole where A = 0.055 m, L = 5 m, V = 72 km/h at 2.5 s (corresponding to a high frequency disturbance). The chosen road profile corresponds to a disturbance satisfying Assumption 2, with δ = 0.5 m2 /s2 . As seen in Fig. 2-3, we can improve the passenger comfort (by minimizing the peak value of the car body acceleration) of the closed-loop system with the proposed method w.r.t the passive open-loop cases (Soft Damper (c = cmin ), Hard Damper (c = cmax ) and Nominal Damper (c = cnom )). Observe that between 2.5 s and 3 s, the control effectively saturates, but the stability is kept. Indeed, during the saturation, the anti-windup acts and the performance does not degrades. Furthermore, it should be noticed that the limits of the suspension travel and the validity for the LPV system are not violated by the trajectory.
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R EFERENCES
Road Profile [m]
0.2
0.1
0
−0.1 0
0.5
1
1.5
2
2.5
3
3.5
4
Road Profile Velocity [m/s]
Time [s] 1 Bound on disturbance amplitude
0.5 0 −0.5
Bound on disturbance amplitude
−1 0
0.5
1
1.5
2
2.5
3
3.5
4
Time [s]
Car Acceleration [m/s2]
Fig. 1.
10
Soft Damper Hard Damper Nominal Damper Proposed Method
5 0 −5 0
Suspension Deflection [m]
Road Profile.
0.5
1
1.5
2
2.5
3
3.5
4
Suspension Deflection Limit (State Constraint)
0.1 0.05 0 −0.05 −0.1
Suspension Deflection Limit (State Constraint)
0
0.5
1
Fig. 2.
1.5
2
Time [s]
2.5
3
3.5
4
Performances Comparison.
θ = z˙def [m/s]
2 Bound on scheduling parameter
1 0 −1
Bound on scheduling parameter
−2 0
0.5
1
1.5
2
2.5
3
3.5
4
Damping coefficient c [Ns/m]
Time [s] 6000
cmax
Saturation bound
Proposed Method
4000 cnom 2000 cmin 0 0
Saturation bound
0.5
1
1.5
2
2.5
3
3.5
4
Time [s]
Fig. 3.
Scheduling parameter and saturation constraint.
VI. C ONCLUSIONS The contribution of the paper is twofold: the proposition of an LMI method to synthesize LPV controllers taking into account input saturation and state constraints; and the application of the method in a semi-active suspension control problem. The simulation results have shown the efficiency of the proposed methodology w.r.t several passive cases. For future work, the application for semi-active suspension control will also be extended further i.e the optimization of comfort, road holding and suspension deflection.
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